Lemma 89.3.1. The functor $F$ (89.3.0.1) is an equivalence.
89.3 Strategy
Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x_1, \ldots , x_ n \in |X|$ be pairwise distinct closed points. For each $i$ we pick an elementary étale neighbourhood $(U_ i, u_ i) \to (X, x_ i)$ as in Decent Spaces, Lemma 68.11.4. This means that $U_ i$ is an affine scheme, $U_ i \to X$ is étale, $u_ i$ is the unique point of $U_ i$ lying over $x_ i$, and $\mathop{\mathrm{Spec}}(\kappa (u_ i)) \to X$ is a monomorphism representing $x_ i$. After shrinking $U_ i$ we may and do assume that for $j \not= i$ there does not exist a point of $U_ i$ mapping to $x_ j$. Observe that $u_ i \in U_ i$ is a closed point.
Denote $\mathcal{C}_{X, \{ x_1, \ldots , x_ n\} }$ the category of morphisms of algebraic spaces $f : Y \to X$ which induce an isomorphism $f^{-1}(X \setminus \{ x_1, \ldots , x_ n\} ) \to X \setminus \{ x_1, \ldots , x_ n\} $. For each $i$ denote $\mathcal{C}_{U_ i, u_ i}$ the category of morphisms of algebraic spaces $g_ i : Y_ i \to U_ i$ which induce an isomorphism $g_ i^{-1}(U_ i \setminus \{ u_ i\} ) \to U_ i \setminus \{ u_ i\} $. Base change defines an functor
89.3.0.1
\begin{equation} \label{spaces-resolve-equation-equivalence} F : \mathcal{C}_{X, \{ x_1, \ldots , x_ n\} } \longrightarrow \mathcal{C}_{U_1, u_1} \times \ldots \times \mathcal{C}_{U_ n, u_ n} \end{equation}
To reduce at least some of the problems in this chapter to the case of schemes we have the following lemma.
Proof. For $n = 1$ this is Limits of Spaces, Lemma 70.19.1. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $g_ i : Y_ i \to U_ i$ are objects of $\mathcal{C}_{U_ i, u_ i}$. Then by the case $n = 1$ we can find $f'_ i : Y'_ i \to X$ which are isomorphisms over $X \setminus \{ x_ i\} $ and whose base change to $U_ i$ is $f_ i$. Then we can set
\[ f : Y = Y'_1 \times _ X \ldots \times _ X Y'_ n \to X \]
This is an object of $\mathcal{C}_{X, \{ x_1, \ldots , x_ n\} }$ whose base change by $U_ i \to X$ recovers $g_ i$. Thus the functor is essentially surjective. We omit the proof of fully faithfulness. $\square$
Lemma 89.3.2. Let $X, x_ i, U_ i \to X, u_ i$ be as in (89.3.0.1). If $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$, then $f$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, if and only if $g_ i$ is so for $i = 1, \ldots , n$.
Proof. Follows from Limits of Spaces, Lemma 70.19.2. $\square$
Lemma 89.3.3. Let $X, x_ i, U_ i \to X, u_ i$ be as in (89.3.0.1). If $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$, then $Y_{x_ i} \cong (Y_ i)_{u_ i}$ as algebraic spaces.
Proof. This is clear because $u_ i \to x_ i$ is an isomorphism. $\square$
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